Suppose you take a positive integer n and add its positive divisors. For example, if n=18, then the sum is 1 + 2 + 3 + 6 + 9 + 18 = 39. In general, when we do this with n one of the following three things happens:
There are infinitely many deficient numbers. For example, pk, with p any prime and k > 0, is deficient. Also if n is any perfect number, and d divides n (where 1 < d < n), then d is deficient.
Deficient and abundant numbers were first so named in Nicomachus' Introductio Arithmetica (c. 100 ad).